Question 1
Let $n, k \ge 2$ be positive integers and let $a_1$, $a_2$, $a_3$, …, $a_k$ be distinct integers in the set $\left\{ 1,2,\dots,n \right\}$ such that $n$ divides $a_i(a_{i+1} - 1)$ for $i = 1,2,\dots,k-1$. Prove that $n$ does not divide $a_k(a_1 - 1)$.